A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

摘要

A new projection method based on radial basis functions (RBFs) is presentedfor discretizing the incompressible unsteady Stokes equations in irregulargeometries. The novelty of the method comes from the application of a newtechnique for computing the Leray-Helmholtz projection of a vector field usinggeneralized interpolation with divergence-free and curl-free RBFs. Unliketraditional projection methods, this new method enables matching bothtangential and normal components of divergence-free vector fields on the domainboundary. This allows incompressibility of the velocity field to be enforcedwithout any time-splitting or pressure boundary conditions. Spatial derivativesare approximated using collocation with global RBFs so that the method onlyrequires samples of the field at (possibly scattered) nodes over the domain.Numerical results are presented demonstrating high-order convergence in bothspace (between 5th and 6th order) and time (up to 4th order) for some modelproblems in two dimensional irregular geometries.